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Theorem reubiia 2543
Description: Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 14-Nov-2004.)
Hypothesis
Ref Expression
reubiia.1  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
reubiia  |-  ( E! x  e.  A  ph  <->  E! x  e.  A  ps )

Proof of Theorem reubiia
StepHypRef Expression
1 reubiia.1 . . . 4  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
21pm5.32i 442 . . 3  |-  ( ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  ps )
)
32eubii 1952 . 2  |-  ( E! x ( x  e.  A  /\  ph )  <->  E! x ( x  e.  A  /\  ps )
)
4 df-reu 2360 . 2  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
5 df-reu 2360 . 2  |-  ( E! x  e.  A  ps  <->  E! x ( x  e.  A  /\  ps )
)
63, 4, 53bitr4i 210 1  |-  ( E! x  e.  A  ph  <->  E! x  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1434   E!weu 1943   E!wreu 2355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-eu 1946  df-reu 2360
This theorem is referenced by:  reubii  2544
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